I used to own this game when it first came out on CD/ rom. However the GOG edition seems unplayable. Whenever my character dies the only options I have are Restart with a level 1 character or Quite Diablo. Everything is lost: all experience, levels etc. Is this typical?

No. You should be able to load your most recent game if you are in single player, or restart in town if you are in multiplayer. For the latter, your items are in the dungeon where you died. If you are lucky or careful, you can get them back. Your experience, attributes, and spells, are retained.

There is pratically no difference between the GOG and the original, in fact it is the original 1.09 exe with some sligh hex edits and a coupple of dll wrappers.
I would say there are two options for what you are experiencing (since you didn't make it clear if you are playing multiplayer or single player),
A: You are playing single player and are forgetting to manually save your progress as you play.
B: Your user does not have write access to the save game folder.

B could be solved by using https://github.com/diasurgical/devilutionX/releases or changing your access levels in Windows.

If none of that helps, please tell us exactly what you did to get to this state. For example, if I was having this problem, I might write:

- Start Diablo from the Windows start menu
- Choose multiplayer
- Choose serial
- Choose my character
- Create a game
- Move at least one item to my inventory
- Go down to the dungeon
- Kill a monster. Note the experience gain.
- Get killed
- Hit escape to open the menu
- The menu offers me: Options, New Game, Restart in Town, Quit Diablo. All of the options are available.
- I choose "Restart in Town"
- I arrive in town reset to level 1, with my inventory and my experience reset to how I started the game

That last one is what I expect happens to you. I've never had that happen to me. My inventory and experience are retained.

As an incidental comment, you can never lose a level 99 character this way. The maximum character level is 50.

Honestly this reads a bit as if OP expects something that is not common with Diablo1?
Like automatic saving or retrieving the equip.
For example:
"If I die at level 3 then I must refight all levels to get back to level 3 with a completely new character."
Yeah, thats the point.
Its not working like Diablo 3.

Just making that clear. Someone who never played Diablo 1 migh interprete some of its mechanics as unintentional compared to the casualized experience of a newer Diablo.
Even Diablo 2 worked differently in some regards.
Maybe that helps figuring out whether there is an actual technical error or a misconception about the game.
As i have to say, from the information given its not clear what it is. OP was not really informative in any way to exclude a misconception.

In single player mode which is story it's by design to do that.

If the player has a saved game prior to dying, then after the character is killed, loading that saved game should be an option and should take the game back to the state it was in when it was saved. I just verified this in a simple single player game with a new character.